Chapter 6: Frequency Domain Anaysis

1. Chapter 6: Frequency Domain Anaysis Instructor: Dr. Cailian Chen

2. Outline Concept Graphical method Analysis • Introduction of Frequency Response • Frequency Response of the Typical Elements • Nyquist and Bode Diagram of Open-loop System • Nyquist-criterion • Frequency Response Based System Analysis • Frequency Response of Closed-loop Systems

3. 6.3.4 Minimum Phase System Definition: A system that has no open-loop zeros or poles in the right hand half plane is called minimum phase system. • Non-minimum phase system: Systems with at least one open-loop pole or zero in the right-hand half of the s plane are known as non-minimum phase system. • A transfer system with delay element is a non-minimum phase system. • For the systems with same magnitude characteristics, the absolute value of phase characteristics of minimum phase systems are minimum. • For minimum phase systems, the magnitude characteristics are uniquely related to the phase characteristics. For non-minimum phase system, there is no such relation.

4. Example 5.8: The frequency characteristics of a minimum phase system is shown as The frequency characteristics of a non-minimum phase system is shown as

5. Bode Diagrams for Minimum and Non-minimum Phase System

6. Example 5.9: Sketch the Bode Diagram of the following system Solution: • It is seen that the magnitude characteristics is same as the inertial element, but the phase characteristics has the extra item –τω . • It implies the phase is lagged very quickly.

7. Example 6.10: The following five systems have same magnitude characteristics.

8. Assume that and . The point at is the geometric intermediate point between and .

9. Remarks： For w=[0,+∞),minimum phase system has the minimum phase change. Moreover, the phase change has the same tendency with magnitude characteristics.(j1(w)) Non-minimum phase system has bigger phase change (eg. j2(w)、j3(w)、j5(w)). Even if the phase change is minimum, the phase change tendency is different from that of magnitude change (eg. j4(w)).

10. 6-4 Nyquist Stability Criterion • Two kinds of frequency domain stability criteria • Nyquist stability criterion and Bode stability criterion • Nyquist stability criterion: based on Nyquist diagram of open-loop system to determine the stability of closed-loop system； • Bode stability criterion: based on Bode diagramof open-loop system to determine the stability of closed-loop system ； • Two method are same in nature • Key point for frequency domain stability criteria： • Determine the stability of closed-loop system based on the frequency response of open-loop system • Moreover, they can be used to determine relative stability.

11. Advantage of Nyquist stability criterion • Geometric criterion: simplicity, graphical method, low computational complexity (Routh stability criterion is an algebraic method） • No need to know the differential equation or transfer function. The frequency responses can be obtained by mathematical modeling or experimental method. • Helpful to understand the concept of relative stability • Mathematical basis for Nyquist stability criterion: Conformal Mapping (Cauchy’s Theorem) 保角映射，又称柯西定理

12. 5.4.1 Conformal Mapping 1. Mapping between s planeand F(s) plane Given a complex function s=σ+jω，F(s) is a complex function，and let F(s)＝U + jV • Assume F(s) is an analytic function (解析函数) except with finite singularity (有限奇点). • The number of zeros of F(s) is same as that of poles. The zeros of F(s) are the poles of the closed-loop system. The poles of F(s) are the poles of the open-loop system.

13. A close contour Γsin s plane which does not go through any singularity There exists the mapping contour ΓFin F(s) plane Mapping between s plane and F(s) plane

14. Remarks： • The ΓFcontour in F(s) plane may go clockwise or anti-clockwise when Γs(s) moves clockwise in s plane. It depends on F(s). • What we are interested in is that Γs(s) encircles the origin a total of N times and the direction rather than the curve shape. • The number N and direction are related to the stability of closed-loop system. （corresponding to the phase change of F(s) ）

15. 2. Phase of F(s) The phase of F(s) can be obtained that Assumption: The contour Γsencircles the zero －z1，and other zeros and poles locate outside of Γs. • When the variable s traversesthe contour Γsin the clockwise direction，the phase change of vector (s+z1)is －2πand that of other vectors are zero. The phase change of F(s) is－2π. • If the contour Γsencircles Z zeros, the mapping ΓF inthe plane of F(s) encircles the origin Z times in the clockwise direction.

16. Contour Γs encircle –z1

17. Similarly，assume Γsencircles P poles. When the variable s traversesthe contour Γsin the clockwise direction, then ΓF inthe plane of F(s) encircles the origin Ptimes in the anti-clockwise direction. 3. Conformal mapping：For a given contour Γs in the s plane that encircles P poles andZ zeros of the function F(s) in a clockwise direction, the resulting contour ΓFin the F(s) plane encircles the origin a total of Ntimesin a clockwise dirction, where N=Z－P How to use the conformal mapping to determine the stability of closed-loop system?

18. 5.4.2 Nyquist Stability 1. Nyquist contour • The sufficient and necessary condition for stability of closed-loop system is that all the zeros of F(s)locate on the left-hand half open s plane. • In order to determine that closed-loop stability，we need to check if there is zero of F(s) locate the whole of the right-hand side of s plane. • Nyquistcontour：encompasses the whole right-hand side of s plane in the clockwise direction. It can encircle all the zeros and poles at the right-hand half side of s plane. • Include two separate segments： • Straight line Cl along the imaginary axis s＝jωwithω=(－∞,＋∞) • Infinite semicircle C2 with infinite radius centered at the origin

19. Nyquist contour without open-loop poles on the imaginary axis

20. 2. Conformal mapping • Based on conformal mapping，when s traverses along the Nyquist contour for one time, the mapping contour ΓF in F(s) plane encircles the origin N = Z－P times in the clockwise direction, where Zand P are respectively the number of zeros and poles in the right-hand half splane. • The sufficient and necessary condition for stability of the closed-loop system is that there is no zeros of F(s), i.e. Z＝0.

21. Analysis： • When s traverses along the Nyquist contour for one time, and the mapping contour ΓF in F(s) plane encircles the origin N = － P times in the clockwise direction, then the closed-loop system is stable. （i.e. P times in the anti-clockwise direction） • If N≠－P，the closed-loop system is unstable. The number of poles in the right-hand half s plane of closed-loop system is Z＝N＋P. • If the open-loop system is stable，i.e. P=0，then the condition for the stability of closed-loop system is: the mapping contour CF does not encircle the origin, i.e.N=0.

22. Characteristic equation of the system • When the contour F(s) encircles the origin，the contour ΓGH of G(s)H(s) equivalently encircles the point (－1，j0). The mapping of Nyquist contour on F(s) plane and G(s)H(s) plane

23. How to draw the mapping contour ΓGH? • Mapping of C1：Let s＝jωand substitute into G(s)H(s)，and then we obtain open-loop frequency characteristics G(jω)H(jω). Draw the Nyquist diagram. Draw the symmetrical part over the real axis for ω=[－0,－∞). • When there are open-loop zeros at the origin, the part from s=-j0 to s=j0 of the Nyquist contour is replaced by a half circle with very tiny radius centered at the origin, i.e. • Draw the open-loop frequency responses for ωfrom－∞ to＋∞，and we can obtain the whole mapping contour ΓGH.

24. Nyquist contour for the open-loop system with poles on the imaginary axis

25. Difference of two kinds of Nyquist contour： • The second kind of Nyquist contour has the center at origin and very tiny radius ε. For ε→0, the area of half circle on the right-hand plane tends to zero. • The zeros and poles of F(s)on the right-hand half s plane are encircled by the Nyquist contour. The open-loop poles at the origin is partitioned into the left-hand half plane. • It is need to know the number of open-loop poles on the left-hand and right-hand half planes.

26. The sufficient and necessary condition for stability of closed-loop system is that：for ωfrom－∞ to＋∞，the open-loop frequency response G(jω)H(jω) encircle the point (－1, j0) for N＝P times in the counterclockwise direction(or N=-P times in the clockwise direction)，where P is the number of open-loop poles on the right-hand half s plane. 3. Nyquist stability criterion • If N≠－P，the closed-loop system is unstable. The number of poles in the right-hand half s plane of closed-loop system is Z＝N＋P. • If the open-loop system is stable，i.e. P=0，then the condition for the stability of closed-loop system is: the mapping contour CGH does not encircle the point (－1, j0), i.e.N=0.

27. Example 5.11: Given the open-loop transfer function Please draw the Nyquist diagram for (1) K=5，(2) K=15，and determine the stability of the closed-loop system.

28. 4、Nyquist stability criterion for systems with open-loop poles on the imaginary axis • Normally with integration elements，i.e. with open-loop poles at the origin of splane • The second kind of Nyquist contour is applicable. • When s traverses along the little half circle from ω=0- to ω=0+，θchanges from －90°to 0°and then further to ＋90°(counterclockwise). The mapping on G(s)H(s)plane is a semicircle with infinite radius. The movement is in the direction of clockwise from ν•90°to 0°and further to －ν•90°. ω：0-→0+； θ：－90°→0°→＋90°； (ω)：＋90ν°→ 0°→－90ν°

29. Example 5.12: Please draw the Nyquist diagram for the open-loop system and determine the stability. Solution：The magnitude and phase characteristics are shown as The Nyquist diagram starts from the third quadrant and ends to the origin in the second quadrant. There exists the intercept with the real axis.

30. It follows from (ω)=－180°that the intercept is A(ω)=1.67 for ω= 1.414. Since there is an open-loop pole at the origin, the mapping of the ε-radius semicircle in the Nyquist contour is infinitesimal semicircle. ω：0-→0+；θ：－90°→0°→＋90°； (ω)：＋90°→ 0°→－90°

31. Nyquist Diagram The Nyquist diagram encircles the critical point (－1, j0). The clockwise encirclement is twice, i.e. N=2. Since P＝0，it yields Z＝N＋P=2. The closed-loop system is unstable, and has two closed-loop poles in the right-hand half s plane.

32. Example 5.13: Draw the Nyquist diagram for the system with open loop transfer function as follows: Determine the stability of the closed-loop system. Solution:The magnitude and phase characteristics are given by Starts from the second quadrant and ends to the origin in the first quadrant.

33. Nyquist Diagram • There are two open-loop poles at the origin. the mapping of the ε-radius semicircle in the Nyquist contour is infinitesimal circle. ω：0-→0+； Θ:－90°→0°→＋90°； (ω):＋180°→0°→－180° • Given P＝0，it can be known from the Nyquist diagram that N = 2. It yields that Z＝P+N=2, thus the system is unstable. There are two closed-loop poles on the right-hand half s plane.

34. Nyquist Diagram In order to calculate the number of encirclement of the critical point, draw a straight line radially outward from the critical point cutting all paths of the Nyquist diagram. 5. How to calculate N? N=N+-N- N+: the number of clockwise encirclements the number of times the path crosses the line in a clockwise sense N-: the number of counterclockwise encirclements the number of times the path crosses the line in the counterclockwise sense

35. 5.4.3 Nyquist Stability Criterion for Bode Diagram Bode diagram based stability criterion is alternative form of Nyquist stability criterion. It uses Bode diagram of open-loop system to determine the stability of closed-loop system. 1. Relationship between Nyquist diagram and Bode diagram • The unity circle centered at the origin0dB line of Bode diagram • Outside of the unity circleL(ω)>0 • Inside of the unity circleL(ω)<0 • Negative real axis of Nyquist diagram－180°line of the log-phase diagram

36. 2. Stability criterion • The sufficient and necessary condition for the stability of closed-loop system: for the range of ωsatisfyingL(ω)≥0，the number of times the phase diagram crosses the －180°line is P (the difference of crossing number in a clockwise and that in the counterclockwise, N=N+-N-),where P is the number of open-loop poles on the right-hand half s plane. • For minimum phase systems, the sufficient and necessary condition for stability is that the difference of clockwise crosses and counterclockwise cross is zero, or (ω) does not cross －180°line.

37. 伯德图 • Example 5.14: The open-loop transfer function is Please apply the Bode diagram based stability criterion to determine the closed-loop stability.

38. 6-5 Relative Stability • Absolute stability → relative stability (how stable) • Time domain, relative stability is measured by parameters such as maximum overshoot and damping ratio. • Frequency domain, the way of measuring relative stability is by how close the Nyquist diagram of G(jω)H(jω) is to the (－1,j0)point. • Relative stability：Suppose that a system has no open-loop poles on the right-hand half s plane, and the closed-loop system is stable. The system is relatively more stable if the path of Nyquist diagram G(jω)H(jω) is far away from (－1,j0). If G(jω)H(jω) crosses the point (－1, j0)，the closed-loop system is critically stable. • Gain and phase margin：express the “closeness”of the path to the critical point. Gain margin (GM) Phase margin (γ)

39. Relationship of system performance between time domain and frequency domain stable Step response stable Step response

40. Relationship of system performance between time domain and frequency domain Marginally stable Step response unstable Step response

41. 1. Phase margin γ • Definition: Phase margin is the amount of pure phase lag that has to be introduced to a system in order to make its frequency response pass through the critical point. • The frequency ωc satisfying A(ω)＝1(i.e. L(ω)=0)is gain crossover frequency (增益剪切角频率). The difference between the phase locus at the gain crossover frequency and the－180°line is phase margin. • If the phase margin is positive, the system is not only stable, but also allow γphase lag to reach the critical stability.

42. 相角裕度和增益裕度

43. For a stable system， (ωc) is above the －180°line. The phase margin is positive (see Fig. (c)). • For an unstable system， (ωc) is below the －180°line. The phase margin is negative (see Fig. (d)) • Correspondingly，γis the phase difference from the crossover point A (of Nyquist diagram and the uinity circle) to the negative real axis. For stable systems,point A is below the negative real axis(see Fig. (a)). For unstable systems,point A is above negative real axis (see Fig. (b)).

44. Phase crossover: A phase crossover on G(jω)H(jω) is a point at which the diagram intersects the negative real axis. Phase crossover frequency: ωg is the frequency at the phase crossover. (ωg)=-180 ° 2. Gain Margin (GM) Definition of Gain Margin: Nyquist diagram Bode diagram

45. 相角裕度和增益裕度

46. For a stable system，L(ωg )in the Bode diagram is below 0dB line. The gain margin is positive (see Fig.(c)). For an unstable system，L(ωg )is above 0dBline. The gain margin is negative (see Fig.(d)). • Fig. (c)shows that the magnitude can be increased by GM dB, i.e. the open-loop gain can be increased by GM times to be marginally stable. • In Nyquist diagram, the distance between the phase crossover point and the origin is 1/Kg ,i.e. . For stable systems, we have Kg>1and GM>0dB (see Fig. (a)). For unstable systems, we have Kg<1and GM<0dB (see Fig. (b)).

47. Remarks： • For an stable minimum phase system, the phase margin is positive and gain margin is bigger than 0dB ； • Being strictly，the relative stability should be given by both phase margin and gain margin. But in engineering, the phase margin is more important. For a good system with satisfactory stability margin, it is desired that

48. The phase margin requirement in engineering implies the slope at the gain crossover frequency should be bigger than －40dB/dec. In system design, we normally set it to be－20dB/dec。 • Given a stable closed-loop system，the relative stability implies the time-domain performance. The bigger phase margin or gain margin, the less overshoot and oscillation and the more damping.

49. Example 5.15: The open-loop transfer function of a unity feedback system is shown as follows: Please obtain the phase margin and gain margin for K1= 10 and K1= 100. Solution: The phase margin can be got from Bode diagram. For K1= 10, Break frequencies are ω1=1，ω2=5. For low frequency part, we have 20lg(K1/5)= 20lg2 = 6dB. Draw Bode diagram as follows:

50. Bode diagram